Basic Theoretical Copula, Empirical Copula, and Various Utility
Functions
Description
This package implements extensive, but select, functions
for copula computations and is used by several other packages
by the author. This particular package provides the lower (W),
upper (M), and product (P) copulas as well as the
product-summation-product copula (PSP). Wrapper functions for
the copula (COP), survival copula (surCOP), dual of a copula
(duCOP), and co-copula (coCOP) are provided. The package has
(level.curvesCOP) for drawing level curves of a given copula,
and this function uses the inverse of a copula (COPinv and a
COPinv2 is provided for completeness). The numerical derivative
for the derivative of a copula is provided by (derCOP and
derCOP2) and the inversion of these functions are by (derCOPinv
and derCOPinv2). The diagonal sections of a copula are
supported by (diagCOP), and sections and derivatives of
sections are supported by (sectionCOP). The inverses of copula
derivatives are important for random variate generation. Random
variate generation for a copula using the conditional
distribution method and the derivative of a copula is provided
by (simCOP and the reduced capability simCOPmicro). For
slightly broader application for purposes of education and
experimentation with copulas, this package also supports the
Plackett copula (PLACKETTcop) because of the general
applicability of this copula. The Plackett copula is
comprehensive, which means that it can attain complete negative
association, independence, and positive association. Plackett
parameter estimation is straightforward with (PLACKETTpar). A
Plackett-specific, random-variate algorithm is by
(PLACKETTsim). Composition of a single copula for two external
parameters is by (composite1COP). Composition of two copulas
through use of two external parameters is provided by
(composite2COP), and the composition of two copulas through the
use of four external parameters is provided by (composite3COP).
Composite copula random variates are generated by
(simcompositeCOP). These compositions generally yield
asymmetric copulas. A data set is provided that contains darts
thrown at the L-comoment space of a Plackett-Plackett
composited compula; these data might be used for experimental
copula estimation by the method of L-comoments. The package
also provides a full suite of functions to numerically compute
measures of association through concordance for a given copula
such as Kendall's Tau (tauCOP), Spearman's Rho (rhoCOP), Gini's
Gamma (giniCOP), and Blomqvist's Beta (blomCOP). The Schweizer
and Wolff's Sigma (wolfCOP) is implemented as a measure of
dependency as opposed the the concordance measures just listed.
Upper and lower tail dependence is computed by numerical limit
convergence by (taildepCOP). A numerical computation of the
logical whether a copula is left-tail decreasing or right-tail
increasing is provided by (isCOP.LTD and isCOP.RTI). The
package supports quantile and median regression through
(qua.regressCOP, qua.regress2COP, med.regressCOP,
med.regress2COP) for a given copula where "2" represents V with
respect to U instead of U with respect to V. The regressions
can be plotted by (qua.regressCOP.draw). **Empirical Copulas**
Empirical copulas are supported by (EMPIRcop) and the
computation of a data frame of the copula for each sample value
is provided by (EMPIRcopdf). The empirical copula functions are
heavily dependent on a simple grid or matrix structure, which
is created by (EMPIRgrid). The derivatives of the grid, which
are the conditional cumulative distribution functions of the
copula sections, are computed by (EMPIRgridder and
EMPIRgridder2). The inverses of the derivatives, which are the
conditional quantile functions of the copula sections, are
computed by (EMPIRgridderinv and EMPIRgridderinv2). Support for
median and quantile regression of the empirical copula are
provided by (EMPIRmed.regress, EMPIRmed.regress2,
EMPIRqua.regress, EMPIRqua.regress2), which use the grids
emanating from (EMPIRgridderinv and EMPIRgridderinv2). Support
for simulation of V using U from an empirical copula is
provided by (EMPIRsim or by EMPIRsimv).